Let $(X,\mu)$ be a space of homogeneous type satisfying $\mu(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain--Morrey--Besov type spaces and Triebel--Lizorkin type spaces associated with the operator $L$. We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, and heat kernel. Atomic and molecular decompositions of these spaces are also given. As an application, we obtain the boundedness of the fractional power of $L$, the spectral multiplier of $L$ on these spaces.

For 1≤p≤q≤∞ and a locally convex space E, we introduce and study the (V∗) subsets of order (p,q) of E and the (V) subsets of order (p,q) of the topological dual E′ of E. Using these sets we define and study (sequential) Pełczyński’s property V∗ of order (p,q), (sequential) Pełczyński’s property V of order (p,q), and Pełczyński’s property (u) of order p in the class of all locally convex spaces. To this end, we also introduce and study several new completeness-type properties, weak barrelledness conditions, Schur-type properties, the Gantmacher property for locally convex spaces, and (q,p)-summing operators between locally convex spaces. Applications to some classical function spaces are given.

Non-collision singularities of the $n$-body problem are initial conditions without global solution, that however do not lead to collision in the limit. The question whether the set of non-collision singularities in the $n$-body problem is improbable,

In this monograph, we present and discuss the many results obtained concerning a famous limit theorem, the local limit theorem (LLT), which has many interfaces, with Number Theory notably, and for which, in spite of considerable efforts, the question concerning conditions of validity of the local limit theorem, has up to now no satisfactory solution. These results mostly concern sufficient conditions for the validity of the LLT and its interesting variant forms: strong LLT, strong LLT with convergence in variation. Quite importantly are necessary conditions, and the results obtained are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary condition, and, almost isolated among the flow of results, Mukhin's necessary and sufficient condition. Extremely useful and instructive are the counterexamples due to Azlarov and Gamkrelidze, as well as necessary and sufficient conditions obtained for a class of random variables, such as Mitalauskas' characterization of the LLT in the strong form for random variables having stable limit distributions. The method of characteristic functions and the Bernoulli part extraction method, are presented and compared. The study of the LLT (old and recent results) consists of three parts: The LLT for sums of independent, identically distributed random variables: Gnedenko's Theorem, Ibragimov and Linnik characterization of the speed of convergence under moments conditions, stronger forms, Galstyan results, strong LLT's with convergence in variation, versions for densities, the case of weighted sums of i.i.d.

In this paper, we consider a two-phase problem for two immiscible, viscous, incompressible fluids in the presence of a uniform gravitational field acting vertically downward in the $N$-dimensional Euclidean space $\mathbf{R}^N$, $N\geq 2$. The two fluids

It is proved that two ternary forms $f$ and $g$ with the same elementary invariants, namely Sylvester signature $\varepsilon $, type $\mathbf{T}$, and $\Omega $ and $\Delta $, belong to the same genus if $aa_{1}$ is a square mod $\Omega $ and $ee_{1}$ is

In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition which generalizes a range of natural "non-freeness conditions" such as the satisfaction of a group law, having nontrivial center or being boundedly simple. As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups to describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results on the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent or classical linear groups.

We investigate the possible ways in which a thick metasymplectic space $\Gamma$, that is, a Lie incidence geometry of type $\mathsf{F_{4,1}}$ (or $\mathsf{F_{4,4}}$), is embedded into the long root geometry $\Delta$ related to any building of type $\maths

In this paper we extensively investigate the class of conditionally positive definite operators, namely operators generating conditionally positive definite sequences. This class itself contains subnormal operators, $2$-and $3$-isometries, complete hyperc

We study isolated points of spaces comprised of homomorphisms between a given ordered AL-algebra and a given unital normed algebra. An ordered AL-algebra is a complex AL-space and simultaneously a Banach algebra such that the positive cone associated with